Many statisticians have had the experience of fitting a linear model with uncorrelated errors, then adding a spatially-correlated error term (random effect) and finding that the estimates of the fixedeffect coefficients have changed substantially. We show that adding a spatially-correlated random effect to a linear model is equivalent to adding a saturated collection of canonical regressors, the coefficients of which are shrunk toward zero, where the spatial map determines both the canonical regressors and the relative extent of the coefficients ’ shrinkage. Adding a spatially-correlated random effect can also be seen as inflating the error variances associated with specific contrasts of the data, where the spatial map determines the contrasts and the extent of error-variance inflation. We show how to restrict the spatial random effect to the orthogonal complement of the fixed effects, which we call restricted spatial regression. We mostly model spatial correlation using an improper conditional auto-regression (ICAR), but briefly show that spatial confounding also arises with socalled geostatistical models and penalized splines, and for the same reason as with the ICAR. We consider five proposed interpretations of spatial confounding and draw implications about what, if anything, one should do about it. For a given problem, the appropriate action depends on whether the spatial random effect is merely as a formal device used to implement spatial smoothing, or a random effect in the traditional sense of, say, Scheffé (1959). For spatial random effects with the former interpretation, restricted spatial regression should be used, while for the latter interpretation this is less clear. In the process, we debunk the common belief that adding a spatially-correlated random effect adjusts fixed effect estimates for spatially-structured missing covariates
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